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Chapter 2: Problem 37

Investigate the following limits. $$\lim _{x \rightarrow 0^{+}}(-10 \cot x)$$

Short Answer

Expert verified
Answer: The limit is \(-10\).

Step by step solution

01

Rewrite the cotangent function

Cotangent function \(\cot x\) can be written as the reciprocal of the tangent function, i.e., \(\cot x = \frac{1}{\tan x}\). Therefore, we have: $$\lim _{x \rightarrow 0^{+}}(-10 \cot x) = \lim _{x \rightarrow 0^{+}}\left(-10 \frac{1}{\tan x}\right)$$
02

Use the limit definition of tangent function

As \(x\) approaches \(0\), we know that \(\lim_{x\to 0}\frac{\sin x}{\cos x}=\lim_{x\to 0}\frac{\sin x}{x}=1\), where we used L'Hopital's rule or geometrical approach for the last limit. Therefore: $$\lim_{x \rightarrow 0^{+}}\tan x=1$$
03

Find the limit of the reciprocal function

Using the property \(\lim_{x\to a}\frac{1}{f(x)}=\frac{1}{\lim_{x\to a}f(x)}\), provided that the original limit exists and is nonzero, we have: $$\lim_{x \rightarrow 0^{+}}\frac{1}{\tan x}=\frac{1}{\lim_{x \rightarrow 0^{+}}\tan x}=\frac{1}{1}=1$$
04

Calculate the given limit

Now, we can plug our result into the given limit: $$\lim _{x \rightarrow 0^{+}}(-10 \frac{1}{\tan x}) = -10 \lim_{x \rightarrow 0^{+}}\frac{1}{\tan x} = -10(1)=-10$$ So, the final answer is: $$\lim _{x \rightarrow 0^{+}}(-10 \cot x) = -10$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit
Understanding limits is fundamental in calculus as it describes how a function behaves as the input approaches a certain value. For example, when examining the limit of a function as it's approached from different sides or directions. Consider the limit presented in the exercise:
  • The notation \( \lim_{x \to 0^+}(-10\cot x) \) tells us to evaluate the behavior of \(-10\cot x\) as \( x \) gets very close to 0 from the positive side.
  • When evaluating limits, sometimes we encounter forms that are indeterminate and require additional methods to solve.
By understanding the core idea of limits, students can analyze how functions behave near points of interest, which is crucial for advanced calculus topics.
Cotangent function
The cotangent function, denoted as \( \cot x \), is a trigonometric function that is the reciprocal of the tangent function. This means:
  • \( \cot x = \frac{1}{\tan x} \), which implies that it is undefined where \( \tan x = 0 \).
  • When approaching limits involving \( \cot x \), it is often helpful to rewrite it in terms of tangent to simplify calculations.
In this exercise, rewriting \( \cot x \) as \( \frac{1}{\tan x} \) allowed for easier manipulation of the limit. Understanding the behavior of \( \cot x \) near points where \( \tan x \) becomes very small or approaches zero is key in evaluating such limits.
L'Hopital's Rule
L'Hopital's rule is a valuable tool used in calculus to evaluate limits that result in indeterminate forms, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). The rule states:
  • If \( \lim_{x \to c} f(x) = 0 \) and \( \lim_{x \to c} g(x) = 0 \), then \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \), provided this limit exists.
  • It can often simplify complex limit expressions, making otherwise tricky problems more straightforward.
  • It was used indirectly here when rewriting the tangent limit, ensuring the resulting \( \lim_{x \to 0^{+}} \tan x = 1 \) was straightforward to calculate.
Utilizing L'Hopital's rule effectively requires knowing when a function results in an indeterminate form, making it an essential technique in solving calculus problems.
Trigonometric limits
Trigonometric limits involve evaluating the behavior of trigonometric functions as they approach a specific value. These limits often recur in calculus problems due to the periodic nature of trigonometry. Key points include:
  • Familiar trigonometric limits are often used as benchmarks, such as \( \lim_{x\to 0} \frac{\sin x}{x} = 1 \) or \( \lim_{x\to 0} \tan x = x \).
  • Trigonometric limits can be resolved by using trigonometric identities to reformat functions into forms where direct calculation is possible.
  • In this exercise, understanding these basic trigonometric limits helped simplify the limit of \( \tan x \), providing a clear path to determine \( \lim _{x \to 0^{+}}(-10 \frac{1}{\tan x}) \).
These key insights allow students to break down complex trigonometric expressions and gauge their behavior near points of interest.

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Most popular questions from this chapter

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence \(\\{2,4,6,8, \ldots\\}\) is specified by the function \(f(n)=2 n\), where \(n=1,2,3, \ldots .\) The limit of such a sequence is \(\lim _{n \rightarrow \infty} f(n)\), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences, or state that the limit does not exist. \(\left\\{0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots .\right\\},\) which is defined by \(f(n)=\frac{n-1}{n},\) for \(n=1,2,3, \ldots\)

Sketch a possible graph of a function \(f\) that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes. $$\lim _{x \rightarrow 0^{+}} f(x)=\infty, \lim _{x \rightarrow 0^{-}} f(x)=-\infty, \lim _{x \rightarrow \infty} f(x)=1$$, $$\lim _{x \rightarrow-\infty} f(x)=-2$$

Evaluate the following limits. $$\lim _{x \rightarrow \pi} \frac{\cos ^{2} x+3 \cos x+2}{\cos x+1}$$

Evaluate the following limits. $$\lim _{x \rightarrow 1^{-}} \frac{x}{\ln x}$$

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$h(x)=\frac{e^{x}}{(x+1)^{3}}$$
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